What is the contra-positive of the following statement? For a, b, `gt 0 , ` If `sqrt(ab) = (1)/(2) ` ( a + b ) then a = b.

Options

A. For a, b `gt ` 0 , if `a ne b ` then `sqrt(ab) ne (1)/(2) ` (a + b)
B. For a, b `lt 0 , ` if `a ne b ` then `sqrt(ab) = (1)/(2) ` ( a + b)
C. for a, b `gt 0 , ` if a = b then `sqrt(ab) = (1)/(2) ` (a + b)
D. For a, b `gt ` 0 , if a `ne ` b then `sqrt(ab) = (1)/(2) ` ( a + b )

Answer

To find the contrapositive of the given statement, we will follow these steps:

### Step 1: Identify the original statement
The original statement is: "If \(\sqrt{ab} = \frac{1}{2}(a + b)\), then \(a = b\)".

### Step 2: Define the statements
Let:
- \(p\): \(\sqrt{ab} = \frac{1}{2}(a + b)\)
- \(q\): \(a = b\)

### Step 3: Write the original statement in symbolic form
The original statement can be expressed as:
\[ p \implies q \]

### Step 4: Find the negation of \(q\)
The negation of \(q\) (which is \(a = b\)) is:
\[ \neg q: a \neq b \]

### Step 5: Find the negation of \(p\)
The negation of \(p\) (which is \(\sqrt{ab} = \frac{1}{2}(a + b)\)) is:
\[ \neg p: \sqrt{ab} \neq \frac{1}{2}(a + b) \]

### Step 6: Write the contrapositive
The contrapositive of the statement \(p \implies q\) is:
\[ \neg q \implies \neg p \]
This translates to:
"If \(a \neq b\), then \(\sqrt{ab} \neq \frac{1}{2}(a + b)\)".

### Final Answer
The contrapositive of the statement is:
"If \(a \neq b\), then \(\sqrt{ab} \neq \frac{1}{2}(a + b)\)".

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